On counting subring-submodules of free modules over finite commutative Frobenius rings
type: Item
instance of: scholarly article
title: On counting subring-submodules of free modules over finite commutative Frobenius rings
zbMATH Open document ID: 1396.13022
author: Alexandre Fotue Tabue
author: E. Martínez-Moro
author: Rama Krishna Bandi
published in: Designs, Codes and Cryptography
publication date: 2018-09-06T00:00:00Z
review text: Let \(R\) be a finite Frobenius commutative ring and \(S\) a finite Galois extension ring of \(R\) of degree \(m\). For each free \(S\)-submodule \(B\) of the free module \(S^{\ell}\) let \(B'=B\cap R^{\ell}\) be the corresponding \(R\)-submodule of \(R^{\ell}\). For positive integers \(k,k'\), the main result of the paper under review is a formula for the number of free \(S\)-submodules \(B\) of the free module \(S^{\ell}\) such that \(S\)-rank of \(B\) is \(k\) and the \(R\)-rank of \(B'\) is \(k'\). The formula is obtained first in the case when \(R\) is a local ring (Theorem 4) and then in the general case (Theorem 6) by decomposing \(R\) as a finite direct product of local Frobenius rings given by the quotiens of \(R\) by adequate powers of the maximal ideals of \(R\). For \({\mathbb F}_q\) a finite field and \({\mathbb F}_{q^m}\) a finite extension of degree \(m\), a by-product of the main result of the paper is a correct formula for the number of distinct \({\mathbb F}_{q^m}\) -linear codes of length \(\ell\), dimension \(k\), and a common fixed subcode over \({\mathbb F}_q\). As the authors point out, this corrects a wrong formula in Theorem 6 of [\textit{B. Lyle}, Linear Algebra Appl. 22, 223--233 (1978; Zbl 0411.15010)].
Mathematics Subject Classification ID: 13M05
Mathematics Subject Classification ID: 16P20
Mathematics Subject Classification ID: 94B05
Mathematics Subject Classification ID: 13C05
Mathematics Subject Classification ID: 16P70
Mathematics Subject Classification ID: 94B25
zbMATH DE Number: 6933688
zbMATH Keywords: Frobenius rings
zbMATH Keywords: linear codes
reviewed by: Felipe Zaldívar
MaRDI profile type: Publication
full work available at URL: s10623-017-0446-1
OpenAlex ID: W2772107888
cites work: Separable algebras over commutative rings
cites work: On the intersection of two subgeometries of \(PG(n,q)\)
cites work: Higher weights for codes over rings
cites work: MDS codes over finite principal ideal rings
cites work: Independence of vectors in codes over rings
cites work: Counting codes over rings
cites work: On separable algebras over a commutative ring
cites work: The dimension of subcode-subfields of shortened generalized Reed-Solomon codes
cites work: A linear-algebra problem from algebraic coding theory
cites work: On trace codes and Galois invariance over finite commutative chain rings
cites work: Q4046212
cites work: Q3041885
DOI: 10.1007/S10623-017-0446-1
Recommended article: Counting codes over rings
Recommended article: Q5744628
Recommended article: On the weight distribution of codes over finite rings
Recommended article: Q4964814
Recommended article: Density of free modules over finite chain rings
Recommended article: Decoding of linear codes over Galois rings
Recommended article: Q4709876
Recommended article: On the Hamming distance of linear codes over a finite chain ring
Recommended article: On trace codes and Galois invariance over finite commutative chain rings
Recommended article: A Griesmer bound for linear codes over finite quasi-Frobenius rings
description: scientific article; zbMATH DE number 6933688